Robust and Adaptive Control System for Regulation of Arterial Gas Pressures by Using Transcutaneous Sensors

Copyright © IF AC Identifica tion and System Parameter Estimation , Beijing, PRC 1988

ROBUST AND ADAPTIVE CONTROL SYSTEM FOR REGULATION OF ARTERIAL GAS PRESSURES BY USING TRANSCUTANEOUS SENSORS A. Sano*, H. Ohmori*, M. Yazawa*,

J.

Xue** and M. Kikuchi***

*Departmfllt of Electrical Engineering, Keio Vllivenit)', 3-14-1 Hiyoshi, Ko/wkZ/ -kll , Yokohama 223 , Japan ** Department of Electrical Engineeri ng, Xi'a n Jiaotong Vllh 'enit)" Xi 'all , PRC ***National Defellce M edical College, Namiki, Tokorowll'a 359, Japall

Abstract: Robust and adaptive control schemes are investigated to regulate arterial oxygen pressure (PaC02) and carbon dioxide pressure (PaC02) independently at their desired levels by automatically adjusting two control inputs of the inspired oxygen concentration (Fi02) and the respiratory frequency (RF) in accordance with continuously monitored transcutaneous oxygen and carbon dioxide pressures (tcP02 and tcPC02), A I inearized input-output model is given through the I inearization at a steady-state level. The parameters and dead time in the I inearized model are identified by using In order to treat multiple models with a fixed dead time and adjustable parameters. with individual differences and variations in the patient and the sensor dynamics and changes of the operating levels, the present control scheme consists of the adaptive control algorithm on a basis of the identified model and the robust controller with The effectivethe Smith predictor to minimize the sensitivity to these variations. ness of the control scheme is investigated in animal experiments using mechanically ventilated dogs Keywords. Biomedical control; Adaptive control; Robust control; tion; Parameter identification.

INTRODUCTION In order to maintain the adequate arteria oxygen pressure Pa02 and carbon dioxide pressure PaC02 in a patient with respiratory failure, it is desired that artificai I respirator should employ a closedloop controller for the regulation by monitoring physiological respiratory state of the patient. Many authors have attempted automatic control of the regulation of PaC02 at predetermined desired level by monitoring the end-tidal C02 pressure (Frumin et al., 1959; Coals et aI., 1973; Mitamura et al., 1971; Kamiyama et al., 1971). In recently developed devices, they adopted the control engineering approach using a reference level explicitly (Chapman et al., 1985; Ohlson et al., 1982) or adaptive control schemes for compensating uncertainties in the control led subject (Sano et al., 1985a) based on the model reference adaptive control theory (Astrom, 1983). In some patients with severe venti lation-perfusion ratio inequal ity or diffusion impairment, the oxygen administration is also needed, because the arterial oxygen pressure (Pa02 ) or the oxygen saturation is below the nor..a I value though PaC02 is normal. Moreover, the control of Pa02 at a desi red level is also necessary to improve not only hypoxia but also hyperoxia which causes sometimes retrolental fibroplasia, venti lation depression and pulmonary collapse. The respiration assisting systems incorporating the oxygen inhalation systems for this purpose have also been studied (Mitamura et al., 1975 Chambille et al . , 1975; Kunke et al., 1976; Kawakali et aI., 1981) . Recently they have tried to apply the optimal control theory (Giard et al., 19

Artificial respira-

85) or the various adaptive control schemes (Sano et al., 1984; Yu et al., 1987) to attain higher control pereforman ce. The present paper presents the robust and adaptive control system for regulating the arterial gas pressures (Pa02 and PaC02) separately at predetermined levels by automatically adjusting two control inputs of the inspired oxygen concentration (Fi02) and the respiratory frequency (RF) which are to be determined in accordance with continuously monitored transcutaneous oxygen pressure (tCP02) and carbon dioxide pressure (tCP02). Since we used the transcutaneous sensors for the c losedloop contro I, we shou Id first construct a I i nearized input-output model including a dead time which is, however, uncertainly changeable according to condition of sensors and skin. Furthermore the model parameters are also variable in accordance with individual differences in respiration desease and alterations of the operating point. Hence, in identification mode, we incorporated multiple models with different dead time to estilated the model parameters as wel I as the dead time. The control scheme consists of an adaptive control algorithm using these identified parameters and a robust controller with the Smith predictor to minimize the sensitivity of variations in the controlled subject. CONTROL SYSTEM CONFIGURATION FOR REGULATION OF ARTERIAL GAS PRESSURES In the proposed control system for assisting respiration for patients with respiratory insuffi1233

A. Sa no et al.

1234

ciency, the arterial gas pressures Pa02 and PaC02 are monitored by transcutaneous sensors of P02 and PC02. Corresponding to these outputs, the microcomputer determines the optimum inputs of the oxygen concentration of inspired gas (Fi02) and the respiratory frequency (RF) on a minute-by-minute basis, and actuates the driver units for changing We can also adopt the the respiratory frequency. end-tidal PC02 or the oxygen saturation as the control led variables in place of tCPC02 and tCP02 respectively, and the minute venti lation quantity as a control input inplaceofRF. INPUT-OUTPUT MODEL OF RESPONSE DYNAMICS OF TRANSCUTANEOUS GAS PRESSURES

d TI' d1 Pa02 (t)

Pa02(t) + bl I Fi 02(t) + bl 2 Ll RF (t) + Cl

d T2' d1 PaC02(t) =

(5)

PaC02(t) + b22 LlRF(t) + C2

(6)

where the coeff i c i ent parameters are given by bll=K

(VT-VD)(Ps-47)RF* I (VT - VD) RF* + V02 (R - I)

bl2 = K V02 (VT - VD)[(R - I) RF* + (Ps -47)] I [(VT - VD) RF* + V02 (R - 1)]2 (VT - VD )(Ps - 47) VC02 b22 = - K2 [(VT - VD) RF* + V02 (R - 1)]2

Linearized Model Mathematical models describing physiological behavior of the respiratory control system in a I iving body, as were studied by Grodins (J967), Mi Ihorn (1965) and others, are rather complex and are not suitable for describing the input-output model of the controlled subjects for the control purpose. Therefore, we shal I consider a simple I inearized model which represents the response dynamics of the arterial oxygen pressure (Pa02) and carbon dioxide pressure (PaC02) for changes of the inspired oxygen concentration (Fi02) and the respiratory frequency (RF). We can approximately describe the response dynamics of the arterial gas pressures for their given initial condition (Sano et al; 1985b), as given by Pa02(t)= Pa02(00) -[Pa02(oo) - Pa0 2(0)]exp(-tlTI)

(I)

PaC02 (t) = PaC02 (00 ) - [PaC02(oo) - PaC02(0)]exp(-tlTI)

(2)

where the each steady-state value is described by

where Cl and C2 are associated with errors in the steady-state of Pa02 and PaC02 respectively, and are regarded zero normally . These equations represent the relationship between the above coefficients and the respiratory physiological quantities. It should be noticed that the model parameters are related with the steady-state operating level of the respiratory frequency RF*, and then they wi I I change according to alterations of the operati ng poi nt. Discrete-Time Input-Output Model of Transcutaneous Gas Pressures Since we use the transcutaneous gas sensors for monitoring the arterial gas pressures, we should consider transport delay between the aorta and periphery, and diffusion from peripheral artery to avascular epithel ium. The transcutaneous sensors have a heater to promote the diffusion, which contributes to reduction of the dead time. Thus, we assume that the above process can be described by a first-order system with the dead time. Finally, combining this time constant with Ti', we can describe an overall input-output I inearized system in the discrete-time form as g(k)=Gp(z)u(k)+GD(Z)d

(4)

where Ps is the barometric pressure, VT the tidal volume of respiration, VD the dead space in lung, R the respirately quotient, V02 the minute oxygen consumption, VC02 the minute production of carbon dioxide, and KI and K2 are the efficiency constants of the gas exchange (0 < KI < 1 and K2> I). It is noticed that the eqns. (3) and (4) include the nonl inearity between the control input RF and arterial gas pressures Pa02 and PaC02 and also involve the product of RF and Fi02, where Fi02 is the other control input. Therefore, we appl ied the linearization technique to (3) and (4) at the normally operating level or steady-state points (Fi02* andRF*). We consider the variations as

and take the linearization in (3) and (4), then we have

(7)

where g (k)=[YI (k),Y2(k»)T, u (k)=[uI (k),U2(k)]T , d=[dl ,d 2]T, and

Gp(z)=

rl I z- L z • PI I 0

G .(,) = [

Z

Z-L - PI I 0

rl 2 Z- L Z - PI I r22 z- L z - P22 0 Z-L Z - P22

where z denotes the time-shifting operator, then we have zy(k)=y(k+l) and z-Iy(k)=y(k-I). The output YI(k) indicates LltCP02, which is the deviation of the transcutaneous oxygen pressure tCP02 (1IHg) from the steady-state tCP02* at the tile instant k S, where S is the salllp ling i nterva I. The output YI (k) is Ll tCPC02 which is also the deviation of the transcutaneous dioxide pressure tcPC02 (uHg) froll tcPC02*. The input UI (k) is LlFi02, which is the deviation of the inspired oxgen concentration Fi02 from Fi02*, and u2(k) is the deviation LlRF frol RF* (cpm). dl and d2 are the uncertain errors relating to the steady-state

Control System for Regulation of Arterial Gas Pressures

operating point. L8 denotes the dead tile existing between the inputs and outputs, which tends to change with the condition of skin and the respiratory desease. ADAPTIVE CONTROL SCHEME WITH ROBUST CONTROLLER The purpose of the controller is to maintain the arterial gas pressures at their desired level as wel I as attain given transient specifications for ensuring quick response dynamics and stabi lity. Adaptive control schemes are required for the following reasons: (a) We should cope with individual differences and changes in respiratory kinetics by adjusting the controller parameters according to the identified model parameters and dead time in order to attain the desired transient specifications. (b) It is rather difficult to identify the model parameters from ordinary step response curves and steady-state values in a case of hypercapnia . Therefore, we should identify the model parameters in an on-I ine manner. (c) A physician sometimes changes the steady-state operating point such as RF* which is related with the model parameters in the I inearized model as given by (5) and (6). Hence we should adjust the controller parameters adaptively in an on-I ine manner according to the alterations of the steadystate operating level. Parameter Identification Output Models

Using Multiple Input·

The parameters and the delay time L are uncertain in the model (7), as lentioned in the previous section. However, it is difficult to identify the model parameters and delay time simultaneously. Hence, in order to estimate the delay L seperately we have taken multiple models with different delay t ille L', where l:iii L';::! D. The output of the each lodel with the delay L' are rewritten from (7) by y, (L') (k+1) =

(J,

Y2 (L' ) (k+ I) =

(J

Norlally we can set d, (L' ) = d2(L') = O. For the sake of silPlicity, the superscript L' is olitted in the followings. The least squares estillates of (J, and (J 2 can be ca I cu I a ted on the bas i s of the past N input-output data in the case of batch processing, as g ;(N)

for i= I and 2

The adequate estiute of L' is deterlllined in an off-I ine lanner by selecting the one from the lultiple lIodels that has the dead time lIIinilizing the next criterion:

IN ( L '

)

=

2

[y ; (k)_i) ;( L')X; (L') (k)] 2 (10)

The purpose of the adaptive control is to achieve a specified transient behavior of the controlled outputs to changes of the reference levels ,u(k), even in the presence of uncertainties in the controlled subject. We can describe the transient specifications in terms of the reference model taking the transfer function GM(Z), as

o D~

L

o In order to coincide the transfer fun ct ion frolil,u to g wi th the reference model GM(Z) exactly, we should design the control input u(k) for attaining the exact model-matching (Wolovich, 1974), as shown in Fig.l, as u

(k)=

Q - , (z)K(z) u

(k)+ Q-' (z)H g (k) +GTM(Z),u(k)

(11)

The condition for the model-matching is given by

= Q (z)[P(z) - G PM(z)RM(Z)]

(12)

where G p( z)=R(z)P-'(z)

Q(z)=diag [Z L ZL] T M(z)=diag [b,z-o .L b2z- 0
G=

[~

r '2 r, , r2 2 _ I r22

1

The controller parameters in K (z) and H are determ i ned f rOla the condition given in (12) as

(9)

where

F ; (N) = • ; (N)/N + P NI g ; (N)= l/J ; (N)/N ' ; (N)=';(N-I)+x ; (N)x ; (N)T l/J ; (N)= l/J ; (N-\)+y ; (N)x ; (N)

N

E E N k=1 i=1

Adaptive Control Schele

(8)

where (J, (l') =(p "fL ' ) , rll (L') , r' 2( L' ), d, (L'»T 9 2 (L ' ) = (P 22 (L ' ), r22 (L ' ) , d2 (L ' ) ) T x, (L') =(y,(k), u,(k -L'), u2(k-L'), 1)T X2(L ') =(Y 2(k), u2(k-L'), I) T

l) ; (N)=F ; (N) -'

The weighting constant PN plays an important role of ilproving the condition number of the covariance latrix F ; (N) and attaining quick and stable convergency of the estillates even in use of slaller nUlber of data. The optimum value of PN can be deterlined by using only the eigenvalues and eigenvectors of F ; (N) and the paraleter estilates (Sano, 1988). The above algorithm can also be rewritten into the on-line recursive form ,

K(z)P(z)+ H R( z )

(L ') T x, (L' ) (k)

2 (L' ) TX 2 (L' ) (k)

1235

L-I E k,,( j)Zi j=O

L-I E k' 2( j)Z j j=O

0

L-I E k22 (j)Z j j=O

K(z)=

1236

A. Sa no et al.

H=

w; (k) = jl ; (k)- gM' ; ; - , (z)F ; (z) c ; (k)

o

(13)

where gM' ; ; (z) = gM ; ; (Z)Z D= b; I (z - a; )

where k" (j) = (a , - p, , )p, ,

( L -, ) - j

F; (z)=S; (z)/(I -S ; (z»

k'2(j)=[k " (j)-k2 2(j)]r , 2!r" k, ,(j)=(a, -p")p,, (l - ' ) - j h, , = (a, , - p" )p, , L Ir, , h, 2 = - (a, - P22 ) P22L r, d (r, , r22) h22 = (a2 - P22)P22 L /r22

S; (z)=(i-f ; )/(z-f;):

lowpass filter

- Predicted output error c ; (k) = y; (k+D) - YM ; (k+D)

As a result, the adaptive control scheme is implem ented by replacing {r ; j , p;; and L'} in (11) and (12) with their identified estimates . Robust Control Scheme The adaptive control can be achieved by combining the model-matching with the parameter identificationy. However, if the parameter estimates converge to constant values and the transfer function from 11 to g becomes a Imost near to the reference model OM(Z), we can stop the identification procedure for the adaptive control and use the modelmatching controller with the obtained fixed paralIeters. In this case, the errors appear in the model-matching and also in the outputs, because the model I ing error is caused by the approximation of the higher order system by the first-order model with the dead time. In order to construct the control system which is insensitive and robust to these model I ing errors and uncertainties in the controlled subject, we sha II introduce the robust contro I scheme wh i ch was developed to control the blood glucose level (Sano et ai, 1987a). However, since the model in this paper includes rather large dead time, the lowpass filter cannot attain wide band. Then we wi II incorporate the Smith predictor to improve the effect and stability of the robust controller, which can be summarized as (See Fig.3):

- Smith predictor y; (k+D)=(I-z-D)[b,/(z-a,)]w ; (k)+y ; (k) (14) The above control can maintain the specified model matching as well as attain the minimum sensitivity to the uncertainties in the control led system . The robust controller in (13) has a simple structure. More general robust controller can be designed so as to guarantee the robust stabil izabi I ity as wel I as the model-matching (the transient specifications), based on the constrained H= method (Ohmori et ai, 1986). The proof that the above robust controller can reduce the sensitivity to variations and disturbances in Op(z) can be made sim i larly as previously reported (Sano et ai, 1987a). Let b. be the error in the mode I-ma tch i ng. If the robust controller is not adopted, the output y; is strongly influenced by b. as y;

=

(15)

;(I+b.)jl ;

gM;

On the other hand, by inducing the robust controller by (13), the input-output relation in (15) can be modified by 1+b.

(16)

Since S; is the lowpass fi Iter with unity gain, it

- Robust contro I Reference mo del gmii (z) A

g mll (z) [

o

,-- - -- - --- -------- -- -

1

0

I I

, I

~----~'~

gm22 ( zl

~

~~------------Y

mi

O:: +D)

r-- - - - - - - - - - - - - - - - - - - - - ;..:-_- _- _:.::-_-::_-:.:_-::-..- - ' I

-

Adapti v e model-mat c hing

"

:

d

'

r-------------------~

:

Wi(k):

" ["1] ,: LJ 2

:

9mi i( z);;

9 ii ( z) m

:

i---r---+---

: (See Fi g . I. i n detail) : IL _________________ ~I

"i ( k)

I

------

-----,

, I I I I I I I

,

I I

I

I IL __ _________ _ _ __ _ _ __ _ _________ __ ______

(

I

~

,

~

ei CK+D)

Gm ( z)

Robus t c o ntro ller

Fig. I

Adaptive model-matching to specified decoupled reference model GM(Z).

:

_____________________________________ J(

Fig. 2 Robust controller incorporating Smith predictor for attaining robust modelma tch i ng.

Control System fo r Regulati on of Arterial Gas Pressures

follows that y;

g"; ; IJ. ; in low frequency range.

1237

the dead tile using the lultiple lodels. Figs. 4, 5 and 6 illustrate the tile courses of the transcutaneous gas pressures controlled by the indicatred control inputs. and cOlpare thel with the nUlerically obtained results using the I inearized lodel with the identified lodel paraleters. In Fig.4, the control purpose is to change only tcPOz frol 801lHg to the new reference level, with keep i ng tcPCOz a t a fixed ref erence Ieve t. In Fig.5, the controller attelpted to achieve quick recovery of tcPCOz frol a high level 45 IIHg to the norlal level 37 I.Hg, without affecting tCP02. The experillental results differed si ightly frol the theoretically calculated results. However, ventilation quantity was increased by rising RF in order to decrease tCPC02 to a nor.al level, whi le the increase of tcPOz caused by the increase of RF could be successfully avoided by adjusting FiOz low. Fig.6 shows the control results of tcPOz and tcPCOz in the case that the each reference level was altered at the sale tile .

RESULTS OF ANIMAL EXPERIMENTS On a basis of the robust and adaptive control algorithls, the licrocolputer (NEC 9801) calculates the two optilul inputs of Fi02 (=Fi02*+uI) and RF (=RF*+u2) and drives the actuator of the oxygen-air lixer and a stepping lotor for adjusting the respiratory frequency RF respectively. As shown in Fig.3, for the purpose of lonitoring Pa02 and PaCOz, thetranscutaneous POz and PCOz sensors are placed with adhesive rings to the chest of a lechanically venti lated dog. The sensor output signals are fedinto the licrocolputer through a 12 bit AID converter. The salpl ing interval was chosen frol ten to thirty seconds_ The control systel treated with dogs of approxilately 10 kg in weight, which were anesthetized by sodiuI pentobarbital. The autonolous respiration was sup~ress­ ed by adlinistering luscle relaxants, and endotracheal intubation was perforled for ventilation.

COlparisons of the proposed adaptive schele with a fixed controller are sUlllarized in Fig.7. In this anilal experilent, we silulated that the loop gain relating to KI was decreased to one half of

Fig.4 shows an exalple of the estimated dead tile chosen frol the lultiple lodels, that linilized the criterion (10). The cOlputer always perforls the paraleter identification and the lonitoring of

..., c:

6

0

tcpo

'M

2

~ Q)

.,

'M

4

~

u

~

Q)

.,> .....'" &

'M

Microcomputer

2

0

2

6

4

8

Delay time L'

Fig.3

Schelatic diagral of the control systel for regulating arterial gas pressures by lonitoring transcutaneous gas pressures .

ZOO

...

co

~ - . -.-.---.

150

Sa.

:::

N

... 0

~

FiOZ

100

RF

...

tdeOz

50

50 "'":.:".:.:-~--.:.:.::::=:::---=-:"'-=::--


u

0

!'"

:::

H

...

'"

0

FiOZ

N

'"

N

0

15

'"

50

'"

0

20

0

Sa.

0

...'"

50

0


FiOZ RF

~

'"

tePCOZ 50

-----------------------------------

:::

0

;..

...

la

15

ZO (min) (b) Simulation result

.. N

0

...~

Fig.5 Results of adaptive control of tcPOz to the reference level without affecting tcPCOz.

!'"

150 FiOZ

N

.

100

...

50

;..

/ .

tdeOz

.

-_:"-:'-.:: .~ .; .;~..; .;~~~ .;,, ~~.~~

0

Sa.

tePOZ

RP

I

..

:::

'"

0

tdeOz 50 ---"'-.:- =2:::~-';·. --':'~=-_7':'::.-

.-.-.

5 N

0

0;: ZO 0 15 (min) (a) Experimental result la

ZOO


0

50

~

0

...u

'1:.

RF


Zo 15 (min) (a) Experimental result


FiOZ

100

u

la

-;;,

tdOl

150

..


00;: 5

Sa.

0

...u 0

u

(1Ilin)

N

lOO

~

N

ZOO

---- -----_.

't

u

100

U

ZOO

u

Sa.

00;: la

tdOz

,! 150

0

...u

.-.- .- .

1i

150

(a) Experimental result

i

zoo co

~

N

0 U

Criterion J versus delay tile L (L=6 is identified).

ZOO tdOz

~

!

Fig.4


'1:.

50

'". 5 N

a

00;:

5

10

15 ZO (min) (b) Simulation result

Fig.6 Results of adaptive control of tcPCOz to the reference level without affecting tcPOz.

!

tdOl

150

e"-

N

0

...u :::

100

.

0' 't

.


F10Z

~

'".

Rp 50

0

tdeOl

-'--=-_._-= - -~------=--5

la

15 ZO (min)

50

5 N

a

00;:

(b) Simulation result

Fig.7 Results of adaptive control of both tcPOz and tcPCOz to each reference I eve I.

1238

A. Sano et af.

200r-----------,

j ~ N

~ •

€ ~

150 .- -- . - .- . - . - . 100 50

j

_._.-

L------::-;;;;--:-j e tcP02 100:; ~

tcPC02 Ry................ .... ..... ...... . ___ .. .. .

.

200r-----------,

'".

~

.::'*·~·::.------,-1-0-2-------------

~

0'-----------:-------'100

~ ...

150

co

.- . - . - . -.-.~ . ----~

tcP° 2

8 100 ~

'[ 100~

F102

tcPCOz .•....... - ....... ___ .• __ _

€ ~

50

..... ,,:::!_ ....................... _..... .

..;,....... "

Rp

...

'".

~

~ N

N

0'------~-----~10Q

(min)

(a)

(min)

......a

!"

."...

!:

!...

....

i\2

o l;:::!:.tF*'-----~- rzz 4 (min)

(b) (c)

Fig. 8 Co.parison of non'adaptive controller and proposed adaptive controller: (a) Non ' adaptive controller with fixed para.eters, (b) Robust and adap' tive controller, and (c) Identified .odel para.eters used in (b) . the nominal value. Hence, the fixed controller could not attain the quick response of the speci· fications, whi le the adaptive controller with the robust loop could adaptively increase the inspired oxygen concentration Fi02 corresponding to the identified .odel para.eters (the esti.ate rll(k) changed low quickly and s.al I gain KI could be detected successfully). CONCLUSIONS The robust control sche.e co.bined with the adapt· ive algorithm has been presented so as to regulate Pa02 and PaC02 seperately at desired levels by adjusting the two control inputs Fi02 and RF according to the .onitored tCP02 and tCPC02. The least squares schele using the .ultiple lodels with different dead ti.e and adjustable para.eters is effective to deterline the linearized first· order input'output .odel including uncertain para' meters and dead time. The adaptive model'matching algorithl included the robust controller with the Slith predictor to linimize the sensitivity to lodel ing errors, variations and disturbances in the controlled subject. The anilal experilent results have clarified that the proposed control algorithl can be easily implemented in an ordinary artificial venti lator. REFERENCES Astrolll, K.J. (1983). Theory and appl ications of adaptive control· survey. Auto.atica, 19, 471 ,486. Chalbi I le, B., H.Guenrad, M.Loncle and D. Bargeton (1975). Alveostat, an alveolar PaC02 and Pa02 control system. J. Appl. Physiol., 39, 837-842 Chap.an, F.W., J.C. Newel I and R.J. Roy (1985). A feedback controller for venti latory therapy. Ann. Bio.ed. Eng., 13. 359·372 . Coals, J.R., W.A. Brown and D.G. Lalpard (1973). COlputer control of respiration and anesthesia Med. i Bi 0 I. Eng., 1\, 262- 267. Frulin, M.J., M.A. Berglan and D.A. Holaday (1975) Carbon dioxide and oxygen blood level with a carbon dioxide controlled artificial respira' tor. Anesthesiology, 20, 313'320. Giard, M.H., F.O. Bertrand, D.Robert and J.Pernier (1985). An algorithl for auto.atic control of 02 and C02 in arterial ventilation. IEEE Trans. Bioled. Eng., BME-32, 658-667. Grodins, F.S., J.Buell and A.J.Bart (1967). Mathe'

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